求取二階矩陣特征向量的一種方法

2024-07-09 03:31:52朱宇航史云鵬

大眾科學 2024年5期

朱宇航史云鵬

摘要：在用冪法求取矩陣的特征值時，將已知向量視為特征向量的線性組合，用矩陣對已知向量做左累乘的迭代運算，相鄰迭代運算向量分量之商就是特征值中絕對值最大者的近似值。通過對迭代過程的無窮小分析可知，在忽略二階無窮小時，近似值與精確值之差呈現等比數列規律，依次進而設計了加速算法，但其結果受二階及更高階無窮小的影響，其與精確值的差值在迭代運算中仍呈現等比數列規律，再次應用前面的加速算法，明顯提高了計算精度，達到了計入二階無窮小的效果，而計入二階無窮小的方程組是無解析解的。

關鍵詞：二階矩陣特征值特征向量加速算法

中圖分類號： O151.21文獻標識碼： A文章編號： 1679-3567（2024）05-0073-03

A Method for Obtaining the Eigenvectors of Second-Order Matrix

ZHU Yuhang1 SHI Yunpeng2

1.China Machinery Technology （Beijing） Vehicle Testing Engineering Research Institute of China， Beijing， 102100 China； 2.China Productivity Center for Machinery Co.， Ltd.， Beijing， 100044 China

Abstract： When using a power method to obtain the eigenvalues of a matrix， a known vector is treated as the linear combination of eigenvectors， the matrix is used to perform an iterative operation of the left multiplication of the known vector， and the quotient of the vector components of adjacent iterative operations is the approximate value of the one with the largest absolute value in the eigenvalues. Through the analysis of infinitesimals in the iteration process， it can be seen that when ignoring second-order infinitesimals， the difference between the approximate value and the exact value presents a geometric progression rule. In turn， an acceleration algorithm is designed， but its result is affected by second-order and higher-order infinitesimals， and the difference between it and the exact value still presents a geometric progression rule in the iteration operation. The previous acceleration algorithm is applied again， the calculation accuracy is significantly improved， achieving the effect of including the second-order infinitesimals， and the system of equations included in the second-order infinitesimals has no analytical solution.

Key Words： Second-order matrix； Eigenvalue； Eigenvector； Accelerated algorithm

動平衡機中需要精確求取具有正弦波的一次諧波系數時，需要對脈沖干擾信號做濾波處理，傳統的低通濾波方法不能在數學上保證求取的一次諧波的精度。因此提出了新的方法：用擬合的數據替代干擾區間的數據進而求取新的一次諧波系數，逐次迭代計算，直至相鄰兩次的迭代值充分接近為止。本文涉及的迭代計算其本質就是用冪法求矩陣的特征值和特征向量。本文只討論比較簡單的二階矩陣。其方法亦可推廣到高階矩陣。